Advanced Matrix Invariants & Academic Operations¶

The algebrax.matrix.academic module provides classical matrix invariants and spectral graph metrics for square sparse dictionary matrices.

Warning

Exact academic functions like determinant(), cofactor(), and adjoint() involve recursive Laplace expansion or \(O (N^5)\) symbolic expansions. They emit a PerformanceWarning for \(N > 10\) and are designed for academic demonstration and exact symbolic verification.

Academic Functions¶

Function	Operation	Description
`determinant(matrix)`	\(\det(A)\)	Recursive Laplace expansion determinant of square sparse matrix.
`inverse(matrix)`	\(A^{-1}\)	Matrix inverse computed via adjugate division \(A^{-1} = \frac{1}{\det(A)} \text{adj}(A)\).
`cofactor(matrix)`	\(C_{i,j}\)	Matrix of cofactors \(C_{i,j} = (-1)^{i+j} \det(M_{i,j})\).
`adjoint(matrix)`	\(\text{adj}(A)\)	Classical adjugate matrix (transpose of cofactor matrix).
`eigen_centrality(matrix)`	\(v\)	Principal eigenvector centrality via power iteration for graph analysis.

Code Example¶

import algebrax as ax

# 1. Square 3x3 Sparse Matrix
A = {
    "0": {"0": 1.0, "1": 2.0, "2": 3.0},
    "1": {"0": 0.0, "1": 1.0, "2": 4.0},
    "2": {"0": 5.0, "1": 6.0, "2": 0.0},
}

# 2. Determinant & Inverse
det_A = ax.matrix.determinant(A)
inv_A = ax.matrix.inverse(A)
print(f"det(A) = {det_A}")
print("A^-1:", inv_A)

# 3. Cofactor & Adjugate
C = ax.matrix.cofactor(A)
adj_A = ax.matrix.adjoint(A)

# 4. Network Eigenvector Centrality
adj_matrix = {
    "Alice": {"Bob": 1.0, "Charlie": 1.0},
    "Bob": {"Alice": 1.0, "David": 1.0},
    "Charlie": {"Alice": 1.0},
    "David": {"Bob": 1.0},
}
centrality = ax.matrix.eigen_centrality(adj_matrix)
print("Eigenvector Centrality:", centrality)